Has anyone ever had success getting the residue out of a western shirt that an iron on patch leaves behind?

You can try Goo Gone but I would test it on a spot you can't see on the shirt first.  I also do a lot of iron on appliques to projects and one way I remove the residue if I have to remove a patch is to take a piece of cotton fabric, place it on the left behind residue then use a hot iron to reheat it and make it stick to the cotton.  I keep putting a clean piece of cotton ironing it until the residue is gone.  Or you can put the patch back on the spot and reheat it to see if it will lift the residue but make sure to lift the patch while it is still hot don't let it cool.  If you use the iron method protect the shirt from iron scorch.  Other options are finger nail polish remover, sometimes alchohol but again test a spot on the shirt you can't see first.

I use acetone or nail polish remover when I have removed crystals on my shirts, which is a bit stronger heat activated glue than a patch. I have been 100% successful on some shirts, but some shirts seem to soak up the adhesive. I have one white 100% cotton shirt that I never have gotten the adhesive out completely (now one of MY shirts to ride in!). I use 15% peroxide to whiten that area and it stays white for a while and then has to be re-treated. Black is actually the hardest color to come right again. I've found a combo of remover, then heat, then remover, several times, seems to work. 

Do test any remover first on the tail of your shirt in case it lifts any color. 

Butane also works on many adhesives, tho it doesn't work on hotfix crystal adhesive.